Brownian Motion

Background

Brownian motion describes the stochastic movement of particles as they travel through space. This type of random movement is often referred to as a random walk, which is typical of particles that diffuse about a 1D 2D or 3D space filled with other particles or barriers. To understand Brownian motion, lets start by characterizing this phenomenon in 2D space.

The mathematical description of this process often includes these terms:

• D - diffusion rate coefficient

equations: D = L²/2d_i

L or λ = step length

d_i = dimensions

units: µm²/s

D is the mean diffusion rate per unit time, often in µm²/s for biological motion on a molecular level. This refers to how fast, on average, the particle moves along its trajectories. This value is often of ultimate interest, particularly for simulating Brownian motion; however under most circumstances, the diffusion rate cannot be observed directly in empirical experiments - this would require the ability to visualize all the microscopic particles and collisions that dictate the particle's movement, and would have to be done on a nanosecond timescale. In the animation on the right, the diffusion rate can actually be quantified directly; but what is often seen when observing particle diffusion through a microscope would more closely resemble this: ^FIG: {{#info: {{{2}}} CLICK AWAY FROM IMAGE TO CLOSE }}. Instead, D is often calculated from the mean squared diffusion (MSD) path of the particle, defined below.

• MSD - mean squared displacement (Δ)

equations: MSD = 2d_i⋅D

units: µm²

MSD is the mean squared displacement of a particle over some time period.

• k - standard deviation of the normal distribution for D

equations: k = √(d_i⋅D)

units: µm

k is the standard deviation of the normal distribution of step lengths that, when randomly sampled, will give rise to a diffusion rate D. This value is useful for simulating Brownian motion for a particular diffusion rate.

Mean Squared Displacement

MEAN SQUARED DISPLACEMENT EQUATIONS

• Δ = 2d*D
• k = √(d * D)
• µⁿ = k * √2 / √π
• λ = √(2d * D) or (2d * D)^0.5
• λᵃ = λ/√2
• Λᵈ = 1/√10

• Δ mean squared displacement (MSD)
• d dimensions
• D diffusion rate
• k σ of sampling distribution that yields D (µ=0)
• µⁿ mean of half normal distribution with σ = k
• λ mean length of the diagonal XY step
• λᵃ mean X & Y step size components of λ
  • 2a² = λ²
  • λ = √2*a
  • a = λ/√2
  • also: λᵃ * .797 = µⁿ
• Λᵈ scalar constant multiplied with k, λ, or λᵃ converting D to 0.D
  • Λᵈ = reciprocal of √10
  • 1.0/√10 = .31623 = 0.D
  • 0.1/√10 = .031623 = 0.0D

• FOR D = 0.5

• Δ = 2.0
• k = 1.0
• µⁿ = 0.7979
• λ = 1.4142
• λᵃ = 1.0

• FOR D = 5.0

• Δ = 20 (/10 = 2)
• k = 3.1623 (* .31623 = 1.0)
• µⁿ = 2.5231
• λ = 4.4721 (* .31623 = 1.4142)
• λᵃ = 3.1623 (* .31623 = 1.0)

• FOR D = 0.3

• Δ = 1.2
• k = 0.7746
• µⁿ = 0.6180
• λ = 1.0954
• λᵃ = 0.7746

• FOR D = 3.0

• Δ = 12 (/10 = 1.2)
• k = 2.4495 (* .31623 = .7746)
• µⁿ = 1.9544
• λ = 3.4641 (* .31623 = 1.0955)
• λᵃ = 2.4495 (* .31623 = 0.7746)

MSD Explained

• m = 2d*D
• k = sqrt(d * D)
• λ = k * √2 / √π
• λ = (2d * D)0.5
• λᵃᵇ = √(2λ²)/2

• m is mean squared displacement
• d is dimensions (2)
• D is diffusion rate
• k is σ of sampling distribution that yields D (µ=0)
• λ is mean of half normal distribution with σ = k
• λ is the mean length of the XY step
• λᵃᵇ is the step size for both X and Y is λᵃᵇ

In statistical mechanics, the mean squared displacement (MSD or average squared displacement) is the most common measure of the spatial extent of random motion; one can think of MSD as the amount of the system "explored" by a random walker.

More Background

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The Probability Density Function (PDF) for a particle in one dimension is found by solving the one-dimensional Diffusion equation. (This equation states that the position probability density diffuses out over time - this is the method used by Einstein to describe a Brownian particle.

As a particle travels, the molecule is jostled by collisions with other molecules which prevent it from following a straight line. If the path is examined in close detail, it will be seen to be a good approximation to a random walk. Mathematically, a random walk is a series of steps, one after another, where each step is taken in a completely random direction from the one before. This kind of path was famously analyzed by Albert Einstein in a study of Brownian motion and he showed that the mean square of the distance traveled by particle following a random walk is proportional to the time elapsed. In two dimensions this relationship can be written as:

r²=4•D•t

In 1D, since both forward and backward steps are equally probable, we come to the surprising conclusion that the probable distance travelled sums up to zero! This is clearly a useless property to calculate. If however, instead of adding the distance of each step we added the square of the distance, we realise that we will always be adding positive quantities to the total. In this case the sum will be some positive number, which grows larger with every step. This obviously gives a better idea about the distance (squared in this case) that a particle moves. If we assume each step happens at regular time intervals, we can easily see how the square distance grows with time, and Einstein showed that it grows linearly with time.

In a molecular system a molecule moves in three dimensions, but the same principle applies. Also, since we have many molecules to consider we can calculate a square displacement for all of them. The average square distance, taken over all molecules, gives us the mean square displacement. This is what makes the mean square displacement (or MSD for short) significant in science: through its relation to diffusion it is a measurable quantity, one which relates directly to the underlying motion of the molecules.

In molecular dynamics the MSD is easily calculated by adding the squares of the distance. The linear (i.e. straight line) dependence of the MSD plot is apparent. If the slope of this plot is taken, the diffusion coefficient D may be readily obtained.

At very short times however, the plot is not linear. This is because the the path a molecule takes will be an approximate straight line until it collides with its neighbour. Only when it starts the collision process will its path start to resemble a random walk. Until it makes that first collision, we may say it moves with approximately constant velocity, which means the distance it travels is proportional to time, and its MSD is therefore proportional to the time squared. Thus at very short time, the MSD resembles a parabola. This is of course a simplification - the collision between molecules is not like the collision between two pebbles, it is not instantaneous in space or time, but is `spread out' a little in both. This means that the behaviour of the MSD at short time is sometimes more complicated than this MSD plot shows.

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Albert Einstein in a study of Brownian motion and he showed that the mean square of the distance traveled by particle following a random walk is proportional to the time elapsed. This relationship can be written as:

r²=2d•D•t

• r² MSD
• d dimensions
• D diffusion coefficient (diffusion rate)
• t time step

The next important equation is the standard deviation of the diffusion rate distribution. It would be more convenient to call this the step size distribution but the two values are not exactly proportional. That equation is given as:

k = sqrt(d * D * t)

• k σ or standard deviation of the diffusion rate distribution
• d dimensions
• D diffusion coefficent
• t time step

Knowing k we can then generate random steps from a normal distribution with a mean µ = 0 and sd σ = k pulling a value from this distribution for both the X and Y dimensions for each step. In Matlab this would look like:

xyds = k * randn(2,Ndots);

This will generate a random walk with a diffusion rate = D and MSD = 2d•D. We might also be interested in the average step size that is being generated from this distribution to create such a diffusion rate. This can be done by taking the mean absolute value of the generated steps:

mStep ≈ mean(mean(abs(xyds),2))

The analytical solution to this problem however is the mean µ of the half normal distribution, with a standard deviation σ = k :

λ = k * √2 / √π

• mS = k*sqrt(2)/sqrt(pi)

However, this value cannot be used as the X and Y step size, if you're expecting to get back to D. What you need is the Pythagorean length of the XY combined step. Einstein solved this too:

λ = (4 * D)^.5;

mSt = sqrt((2*ld^2))/2;

I'm simulating Brownian motion. Each particle (dot) in the simulation take a random size step in a random direction. This is in 2D. Each iteration, a dot gets passed 2 values, a random step size for the X direction and Y direction. Voila! Brownian motion.

Now to characterize this motion. This wasn't an easy task, until this really smart dude came along and provided a few theorems. Albert Einstein showed that the mean square of the distance traveled by particle following a random walk is proportional to the time elapsed. This relationship can be written as (and hopefully the unicode looks like r^2 and not r2.. I won't put integers directly after variables):

• r² = 2d*D*t

• r² = mean squared displacement MSD (µm²)
• d = number of dimensions
• D = diffusion coefficient (µm²/s)
• t = time elapsed between each measurement (1 s)

Why is the mean squared displacement good to know? Well, creating Brownian motion is easy (like I mentioned above), but simulating Brownian motion based on real-life observations is not so easy. This is because we can't directly observe the step size (the distance traveled between particle collisions), but we can observe the MSD. Ultimately I'd like to estimate the average step size, but having a good estimate of the diffusion rate (D) would suffice for making a simulation based on real-life observations.

Say we're observing a particle through a microscope diffusing along a flat surface (d=2), and each second (t = 1) we write down its displacement (for both X and Y), squaring each value (**see footnote). When we're done, we can calculate the mean of these values and find that MSD=8 µm². We can now find D using the equation above.

• 8 µm² = (2*2) * D µm²/s * 1 s
• 8 µm²/s = 4 * D µm²/s
• 2 µm²/s = D µm²/s

Sweet, we have the diffusion rate. We can now use this value to make a simulation of our empirical observations.... But how? The simulation needs random step sizes, not a diffusion rate. Einstein figured this out too. He determined that the standard deviation (σ=k) of the distribution of random steps is:

k = sqrt(d * D * t)

From our observations:

k = sqrt(2*2*1) = 2

Now all we have to do is pull random values from a normal distribution with µ=0 σ=2 for each X and Y step size. If I do that and run a simulation - lets do 100 steps for 100 particles. boom:

Sweet, it worked! D = 2.0

We could stop there. But we're curious... What was the average step size it was creating to get that D? So we make a subroutine to track that information. We find that the average of the absolute values of the step sizes for each iteration are coming out one by one:

1.48, 1.61, 1.56, 1.51, 1.68

The average of all those values is around 1.59 Why? We can take a more analytical approach. Since we are using the absolute value of each step to calculate the mean step size, we are essentially sampling from a half normal distribution. The mean of a half normal distribution is:

µ = σ * sqrt(2) / sqrt(pi)

Our standard deviation was 2.

2*sqrt(2) / sqrt(pi) = 1.5958

Nice, that's pretty much what we got (1.59).

So, to recap, we started with a diffusion rate D=2, which, using Einstein's equation, helped us determine we needed to generate a bunch of steps from a distribution µ=0 σ=2 which resulted in a bunch of steps that were +/- 1.59 on average. Then as a check, we took the mean square of all 100 steps for each particle and calculated D which turned out to be 2.02 so everything worked as planned.

We could stop there, but will we? Of course not. Let's do one last check, to see if we completely understand what's going on. Say we don't know D or MSD. In fact, we don't have any real-life empirical information to go on. Perhaps all we want to do is generate some random walks, and if we're interested, we can use those walks to calculate the diffusion coefficient (D). Let's make 100 particles take 100 steps with a step size of 1.5958 (randomly negative or positive), aka the average step size from our experiment above.

Before we do this, any guess what D is gunna turn out to be?

D = 1.27

What the fuck!!

Duh! This value cannot be used as the X and Y step size, if you're expecting to get back to D. What you need is the Pythagorean length of the XY combined step. Einstein solved this too. But first, a recap:

• m = 2d*D
• k = sqrt(d * D)
• λ = k * √2 / √π
• λ = (2d * D)0.5

• m is mean squared displacement
• d is dimensions (2)
• D is diffusion rate
• k is σ of sampling distribution that yields D (µ=0)
• λ is mean of half normal distribution with σ = k
• λ is the mean length of the XY step

See our new equation there: λ = (2d * D). Now, all we have to do to get the X and Y step length values, is to calculate what two values equals λ when using the Pythagorean theorem:

• a²+b²=λ²

since 'a' and 'b' can be the same value, lets just use:

• 2a² = λ²

and solve for 'a'

• λᵃᵇ = √(2λ²)/2

so the step size for both X and Y is λᵃᵇ

Image Gallery

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File:Brownian4.ogv

Notes

Finding the MSD of the simulation is (a trivial task) using a Matlab toolbox (see video below). The target extrasynaptic MSD was 0.1 ^µm²⁄_s (from Choquet ^FIG: {{#info: {{{2}}} CLICK AWAY FROM IMAGE TO CLOSE }}). Scaling the model PSD size and distance between synapses to real-world values based on the target MSD is not a trivial task, and may actually be related only indirectly. Just because a particle moves quickly, does not mean it moves from A to B quickly - it depends on how much stuff there is to collide with. Linear estimates have shown that glycine receptor movements along dendrites at a speed of 1–2 µm/min (.016-.008 µm/s) ^FIG: {{#info: {{{2}}} CLICK AWAY FROM IMAGE TO CLOSE }}

First, the randomly generated step-size was scaled to produce an MSD of 0.1 ^units²⁄_step using the Matlab Brownian motion toolbox. Second, the dimensions of the model need to be scaled. It was found that an XY random step-size of µ=0.4 (σ=.2) units produced an MSE ≈ 0.1 ^units²⁄_step. Next, the arbitrary 0.4 units need to be given meaning...

Given movements along dendrites at a speed of 1–2 µm/min (.016-.008 µm/s) observation, we can use a derivation of the MSD equation to help scale the model. The root-mean-square distance after N unit steps, with a step length of (L) is:

• d = L*sqrt(N)

In order to travel a distance d, N steps are required by this equation

• N = (d/L)²
• N = (1000 nm /400 nm)² = 6.25
• N = (2000 nm /400 nm)² = 25
• N = (4000 nm /400 nm)² = 100

• Lets say a particle is traveling linearly at a rate of 1 µm / 100 s
• And takes between 6-25 steps to cover 1-2 µm.

here are several good resources for explaining and calculating MSD:

• wolfram alpha
• Democritus
• Matlab toolbox

VIDEO

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MEAN SQUARED DISPLACEMENT

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Brownian Motion Mean Squared Displacement

• The goal of this calculation is to relate the simulated particle diffusion to real world values, namely velocity.
• Particle velocity will be a function of MSD x units ²⁄s which scales on space (units) and time (s) parameters.
• Space and time in the model are defined arbitrarily as Step_Size and Step where each Step a particle moves a distance randomly chosen from a normal distribution (µ=1,σ=.2)
• a step size of 1 unit/step will produce a brownian motion MSD of ~0.52 ±0.2 units ²/s
• empirical observations show that reasonable values for MSD are:
  • PSD 0.01 µm ²/s
  • synaptic 0.05 µm ²/s
  • extrasynaptic 0.1 µm ²/s

• given an MSD of 0.52 ±0.2 units ²/s at the current parameters: 1 step = 1 unit (at µ=1,σ=.2), the model will need to be scaled such that particles move at an extrasynaptic rate of 0.1 µm ²/s.
• spines are on average 1 to 10 µm apart, if the model is comparing two spines 1 µm apart, they should be separated by 5 units of model space. This is because the current particle diffusion rate of the model is .5 µm ²/s and the empirical MSD is .1 µm ²/s

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to make 0.1 ^units²⁄_step ≈ 0.1 ^µm²⁄_s. It was found that an XY random step-size of µ=0.4 (σ=.2) units produced an MSE ≈ 0.1 ^units²⁄_step. Then, the arbitrary 0.5 units were given meaning (converted to 0.5 µm) by scaling the model according to real-world values (see below) by making 1 unit = 1 µm; as a convention, a subunit will be 1/10th of a unit, thus 1 subunit = 0.1 µm). The PSD areas were set to 3-subunits (.3 µm) square, 20 subunits (2 µm) apart, within a rectangular field 20 subunits (2 µm) wide and 60 subunits (6 µm) long. Given these scaled dimensions where 10 subunits ≈ 1 µm, a particle with an XY step-size of 0.5 units moving in a straight line, could theoretically go from PSD1 to PSD2 in 4 steps (obviously given the simulated particles are moving with Brownian motion, this lower-bound would be extremely rare).

Articles

Michalet • 2010 • Phys Rev E Stat Nonlin Soft Matter Phys - PDF

Mean Square Displacement Analysis of Single-Particle Trajectories with Localization Error: Brownian Motion in Isotropic Medium

Expand to view experiment summary

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We examine the capability of mean square displacement analysis to extract reliable values of the diffusion coefficient D of single particle undergoing Brownian motion in an isotropic medium in the presence of localization uncertainty. The theoretical results, supported by simulations, show that a simple unweighted least square fit of the MSD curve can provide the best estimate of D provided an optimal number of MSD points is used for the fit. We discuss the practical implications of these results for data analysis in single-particle tracking experiments.

Czondor Choquet • 2012 • PNAS - PDF

Unified quantitative model of AMPA receptor trafficking at synapses

Expand to view experiment summary

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^FIG: {{#info: Error creating thumbnail: File missing(A) Schematic diagramof themodel. Kinetic parameters include Dout (extrasynaptic diffusion), Din (synaptic diffusion), DPSD (diffusion coefficient of the PSD), kon (AMPAR/ scaffold binding rate), koff (AMPAR/scaffold dissociation rate), and kendo (endocytosis rate). (B) Simulated trajectory (50 s). The synapse is in green, the PSD in red, and dendrite borders in gray. Geometric parameters are: a (synapse spacing), b (synapse width), c (PSD width), w (dendrite width). (C) High-magnification 30-s trajectory of an AMPAR-bound Qdot on the surface of a DIV 9 neuron. and simulated (plain curves) AMPAR median diffusion coefficients obtained at different neuronal ages (DIV 4–15) or varying synaptic spacing (0.75–30 μm), respectively, were plotted against synapse density. The simulated curves were computed for different kon values, keeping koff: 0.1 s−1. (F) Interquartile distributions of AMPAR diffusion coefficients from experiments (black) and simulations (green). Neuroligin-1 expression, which doubled the number of Homer1c-positive puncta, was mimicked by a decrease in synapse spacing (a: 1 μm). Overexpressing PSD-95 was modeled by enhancing kon (2.5 s−1) to mimic an increase in PSD binding sites, plus an increase in PSD size (c: 0.4 μm). CLICK AWAY FROM IMAGE TO CLOSE }}

Dendrite

• length: 10 - 60 µm
• width: 2 µm
• D.out: 0.05 - 0.3 µm²/s

PSD

• length: 0.3 µm
• width: 0.3 µm
• distance: 1 - 30 µm apart

PSA (perisynaptic area PSD pad)

• length: 0.3 µm
• width: 0.3 µm
• D.in: 0.03 µm²/s

Total synaptic area: .6 x .6 µm In some cases, we varied the size of the synapse and the PSD, keeping a constant ratio of 2 between them (b: 0.1–1.6 μm; c: 0.05–0.8 μm).

AMPARs bound to the PSD were allowed to diffuse at a very low diffusion coefficient DPSD: 0.0002 μm2/s, reflecting the intrinsic movement or morphing of the PSD. We neglected the turnover of individual scaffold molecules, typically much longer than the AMPAR diffusion time scale